The Variation of the Uncentered Maximal Operator with respect to Cubes

TL;DR

This paper proves that the variation of the uncentered maximal function with respect to cubes is bounded by the original function's variation times a constant, extending to certain collections of sets.

Contribution

It establishes a bound on the variation of maximal functions for functions of bounded variation with respect to cubes and more general sets satisfying specific geometric conditions.

Findings

Variation of maximal function bounded by original variation times a constant

Results extend to collections of sets satisfying inner cone star and tiling properties

Applicable to uncentered maximal operators with respect to cubes and certain other sets

Abstract

We consider the maximal operator with respect to uncentered cubes on Euclidean space with arbitrary dimension. We prove that for any function with bounded variation, the variation of its maximal function is bounded by the variation of the function times a dimensional constant. We also prove the corresponding result for maximal operators with respect to collections of more general sets than cubes. The sets are required to satisfy a certain inner cone star condition and in addition the collection must enjoy a tiling property which for example the collection of all cubes does enjoy and the collection of all Euclidean balls does not.